Abstract

We analyze the far field and near field diffraction pattern produced by an amplitude grating whose strips present rough edges. Due to the stochastic nature of the edges a statistical approach is performed. The grating with rough edges is not purely periodic, although it still divides the incident beam in diffracted orders. The intensity of each diffraction order is modified by the statistical properties of the irregular edges and it strongly decreases when roughness increases except for the zero-th diffraction order. This decreasing firstly affects to the higher orders. Then, it is possible to obtain an amplitude binary grating with only diffraction orders -1, 0 and +1. On the other hand, numerical simulations based on Rayleigh-Sommerfeld approach have been used for the case of near field. They show that the edges of the self-images are smoother than the edges of the grating. Finally, we fabricate gratings with rough edges and an experimental verification of the results is performed.

© 2008 Optical Society of America

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References

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  1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1980).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. E. G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, New York, 1997).
  4. M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
    [CrossRef]
  5. S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
    [CrossRef] [PubMed]
  6. C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 2000).
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    [CrossRef]
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    [CrossRef]
  10. G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
    [CrossRef]
  11. F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, "Talbot effect with rough reflection gratings," Appl. Opt. 46, 3668- 3673 (2007)
    [CrossRef] [PubMed]
  12. L.M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, "Talbot effect in metallic gratings under Gaussian illumination," Opt. Commun. 278, 23-27 (2007).
    [CrossRef]
  13. L. M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, "Far field of gratings with rough strips," J. Opt. Soc. Am. A 25, 828-833 (2008).
    [CrossRef]
  14. W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).
  15. K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989).
    [CrossRef]
  16. N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
    [CrossRef]
  17. Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kind of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005)
    [CrossRef]
  18. P. P. Naulleau and G. M. Gallatin, "Line-edge roughness transfer function and its application to determining mask effects in EUV resist characterization," Appl. Opt. 42, 3390-3397 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  20. V. A. Doroshenko, "Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips," Telecommunications and Radio Engineering 57, 65-72 (2002)
  21. M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998)
    [CrossRef]
  22. F. Shen and A. Wang, "Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula," Appl. Opt. 45, 1102-1110 (2006)
    [CrossRef] [PubMed]
  23. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House Norwood, 1987).

2008 (1)

2007 (2)

F. J. Torcal-Milla, L. M. Sanchez-Brea, and E. Bernabeu, "Talbot effect with rough reflection gratings," Appl. Opt. 46, 3668- 3673 (2007)
[CrossRef] [PubMed]

L.M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, "Talbot effect in metallic gratings under Gaussian illumination," Opt. Commun. 278, 23-27 (2007).
[CrossRef]

2006 (2)

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

F. Shen and A. Wang, "Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula," Appl. Opt. 45, 1102-1110 (2006)
[CrossRef] [PubMed]

2005 (2)

Y. Lu, C. Zhou, and H. Luo, "Talbot effect of a grating with different kind of flaws," J. Opt. Soc. Am. A 22, 2662-2667 (2005)
[CrossRef]

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

V. A. Doroshenko, "Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips," Telecommunications and Radio Engineering 57, 65-72 (2002)

2001 (1)

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
[CrossRef]

2000 (1)

N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

1999 (2)

1998 (1)

M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998)
[CrossRef]

1994 (1)

1989 (1)

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989).
[CrossRef]

1836 (1)

W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

Bernabeu, E.

Borghi, R.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
[CrossRef]

Deshpande, A. J.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Doroshenko, V. A.

V. A. Doroshenko, "Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips," Telecommunications and Radio Engineering 57, 65-72 (2002)

Gallatin, G. M.

Glazov, M. V.

M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998)
[CrossRef]

Gori, F.

Guérineau, N.

N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

Harchaoui, B.

N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Lu, Y.

Luo, H.

Michel, T. R.

Mondello, A.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
[CrossRef]

Mueller, G.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Naulleau, P. P.

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989).
[CrossRef]

Piquero, G.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
[CrossRef]

Primot, J.

N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

Quetschke, V.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Rashkeev, S. N.

M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998)
[CrossRef]

Reitze, D. H.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Sanchez-Brea, L. M.

Sanchez-Brea, L.M.

L.M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, "Talbot effect in metallic gratings under Gaussian illumination," Opt. Commun. 278, 23-27 (2007).
[CrossRef]

Santarsiero, M.

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
[CrossRef]

Shen, F.

Someda, C. G.

Talbot, W. H. F.

W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

Tanner, D. B.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Torcal-Milla, F. J.

Wang, A.

White, K. R.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Whiting, B. F.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Wise, S.

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Zhou, C.

Appl. Opt. (3)

Appl. Phys. B (1)

M. V. Glazov., and S. N. Rashkeev, "Light scattering from rough surfaces with superimposed periodic structures," Appl. Phys. B 66, 217-223 (1998)
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

N. Guérineau, B. Harchaoui, and J. Primot, "Talbot effect re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, "Far field of beams generated by quasi-homogeneous sources passing through polarization gratings," Opt. Commun. 195, 339-350 (2001)
[CrossRef]

L.M. Sanchez-Brea, F. J. Torcal-Milla, and E. Bernabeu, "Talbot effect in metallic gratings under Gaussian illumination," Opt. Commun. 278, 23-27 (2007).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. (1)

W. H. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 401-407 (1836).

Phys. Rev. E (1)

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

S. Wise, V. Quetschke, A. J. Deshpande, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, "Phase effect in the diffraction of light: beyond the grating equation," Phys. Rev. Lett. 95, 013901 (2005).
[CrossRef] [PubMed]

Prog. Opt. (1)

K. Patorski, "The self-imaging phenomenon and its applications," Prog. Opt. 27, 1-108 (1989).
[CrossRef]

Telecommunications and Radio Engineering (1)

V. A. Doroshenko, "Singular integral equations in the problem of wave diffraction by a grating of imperfect flat irregular strips," Telecommunications and Radio Engineering 57, 65-72 (2002)

Other (6)

M. Born, and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1980).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

E. G. Loewen and E. Popov, Diffraction gratings and applications (Marcel Dekker, New York, 1997).

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House Norwood, 1987).

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, New York, 2000).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (1040 KB)     
» Media 2: MOV (1856 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Example of the grating analyzed in this work.

Fig. 2.
Fig. 2.

Diffraction orders intensity for different values of the roughness parameters, when, p=20µm, λ=0.68µm T=50µm and L=50µm:σ→0 (solid line), b) σ=0.2µm (dot line), c), σ=0.4µm(dash line). The order 3 disappears for σ=0.2µm and σ=0.4µm.

Fig. 3.
Fig. 3.

Perfect grating and first three self-images obtained using the Rayleigh-Sommerfeld approach. The period of the grating is p=20µm and the wavelength is λ=0.68µm.

Fig. 4.
Fig. 4.

Grating and first three self-images obtained using the Rayleigh-Sommerfeld approach for the same situation of Fig. 3 when the roughness parameters are T=0.1µm, σ=.25µm.

Fig. 5.
Fig. 5.

Grating and first three self-images obtained using the Rayleigh-Sommerfeld approach for the same situation of Fig. 3 when the roughness parameters are T=1µm, σ=1µm.

Fig. 6.
Fig. 6.

Comparison of the average profiles for the first three self-images for two different roughness levels a) for the parameters of Fig. 4 (b) for the parameters of Fig. 5. Grating with rough edges (solid line), perfect grating, Fig. 3 (dash line).

Fig. 7.
Fig. 7.

Fractional self-images for a grating with period p=20µm, the roughness parameters are T=1µm and σ=1µm, and the wavelength is λ=0.68µm for positions a) z=zr /4,b) z=zr /3,c) z=zr/1 and d) average profiles for this fractional self-images. (Media 1).

Fig. 8.
Fig. 8.

Average for the first four self-images obtained using the Rayleigh-Sommerfeld approach. The number of samples was 100, the period of the grating is p=20µm and the roughness parameters are T=1µm and σ=1µm. The wavelength of the incident beam is λ=0.65 µm.

Fig. 9.
Fig. 9.

Optical image of the manufactured diffraction grating with rough edges and experimental first three self-images. The period of the grating is p=100µm, the wavelength is λ=0.65µm. The roughness parameters used are T=50µm and σ=5µm The images are captured with a CMOS camera whose pixel size is 6µm×6µm and a 30×microscope objective. (Media 2).

Fig. 10.
Fig. 10.

(a) Mean profile of the image and self-images shown in Fig. 9 (Media 2) and (b) experimental self image for n=5. The intensity distribution is very smooth. The fluctuations at x=500µm are due to a dust particle in the optics that we could not eliminate.

Tables (1)

Tables Icon

Table 1. Coefficients arough j for the first five diffraction orders of a grating, measured as aj exp[-(j2πσ/p)2/2], for several values of σ/p defined accirding to (10).

Equations (25)

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t ( ξ , η ) = n = ( f n ( ξ ) , g n ( ξ ) ) ,
U ( x , y ) = e ik ( z + x 2 + y 2 2 z ) i λ z t ( ξ , η ) U i ( ξ , η ) e i k z ( x ξ + y η ) d ξ d η ,
I ( x , y ) = U ( x , y ) 2 = U 0 2 ( λ z ) 2 n , n ' = N 2 N 2 L 2 L 2 d ξ L 2 L 2 d ξ
g n ( ξ ) + ( 4 n + 1 ) p 4 f n ( ξ ) + ( 4 n 1 ) p 4 d η g n ( ξ ) + ( 4 n + 1 ) p 4 f n ( ξ ) + ( 4 n 1 ) p 4 e i k z [ x ( ξ ξ ) + y ( η η ) ] d η
I ( x , y ) = U 0 2 ( 2 π z θ y ) 2 n , n L 2 L 2 d ξ L 2 L 2 d ξ e i k θ x ( ξ ξ )
{ e i k θ y p ( n n ) e i k θ y [ f n ( ξ ) f n ( ξ ) ] + e i k θ y p ( n n ) e i k θ y [ g n ( ξ ) g n ( ξ ) ] e i k θ y p 2 [ 2 ( n n ) + 1 ] e i k θ y [ f n ( ξ ) g n ( ξ ) ] e i k θ y p 2 [ 2 ( n n ) 1 ] e i k θ y [ g n ( ξ ) f n ( ξ ) ] } ,
e i α f n ( ξ ) = e i α g n ( ξ ) = e ( α σ ) 2 2 ,
e i α [ f n ( ξ ) g n ( ξ ) ] = e ( α σ ) 2 ,
e i α [ f n ( ξ ) f n ( ζ ) ] = e i α [ g n ( ξ ) g n ( ξ ) ] = e ( α σ ) 2 m = 0 ( α σ ) 2 m m ! e m ( ξ ξ ) 2 T 2 ,
e i α [ f n ( ξ ) f n ( ξ ) ] = e i α [ g n ( ξ ) g n ( ξ ) ] = e ( α σ ) 2 , ( n n ) .
I ( x , y ) = 2 U 0 2 ( 2 π z θ y ) 2 { [ 1 cos ( k θ y p 2 ) ] e ( k θ y σ ) 2 n , n e i k θ y p ( n n ) L 2 L 2 L 2 L 2 e i k θ x ( ξ ξ ) d ξ d ξ
N e ( k θ y σ ) 2 L 2 L 2 L 2 L 2 e i k θ x ( ξ ξ ) d ξ d ξ
+ N e ( k θ y σ ) 2 m = 0 ( k θ y σ ) 2 m ! L 2 L 2 L 2 L 2 e i k θ x ( ξ ξ ) e m ( ξ ξ ' ) 2 T 2 d ξ d ξ } ,
I ( x , y ) = U 0 2 L 2 p 2 ( N + 1 ) 2 ( 2 λ z ) 2 e ( k θ y σ ) 2 sinc 2 ( k θ y p 4 ) sinc 2 ( k θ x L 2 )
j sinc 2 [ π p λ ( N + 1 ) ( θ y j λ p ) ]
+ 2 U 0 2 NLT π σ 2 ( λ z ) 2 e ( k θ y σ ) 2 m = 1 ( k θ y σ ) 2 ( m 1 ) m ! m e ( k T θ x ) 2 4 m .
I ( x , y ) = U 0 2 L 2 p 2 ( N + 1 ) 2 ( 2 λ z ) 2 sinc 2 ( k θ y p 4 ) sinc 2 ( k θ x L 2 ) j sinc 2 [ π p λ ( N + 1 ) ( θ y j λ p ) ] ,
I ( x , y ) ¯ = sinc 2 ( k θ x L 2 ) j a j 2 e ( j 2 π σ p ) 2 sinc 2 [ π p λ ( N + 1 ) ( θ y j λ p ) ]
+ 8 π T σ 2 NL p 2 e ( k θ y σ ) 2 m = 1 ( k θ y σ ) 2 ( m 1 ) m ! m e ( k T θ x ) 2 4 m .
a j rough = a j perfect exp [ ( j 2 π σ p ) 2 2 ] .
e i α [ f n ( ξ ) f n ( ξ ) ] = e i α [ g n ( ξ ) g n ( ξ ) ] = exp [ ( ξ ξ ) 2 T F 2 ] ,
I ( x , y ) ¯ = sinc 2 ( k θ x L 2 ) j a j 2 e ( j 2 π σ p ) 2 sinc 2 [ π p λ ( N + 1 ) ( θ y j λ p ) ]
+ 8 T F π σ 2 NL p 2 e k 2 [ ( θ y σ ) 2 + ( θ x T F 2 ) 2 ] .
e i α [ f n ( ξ ) f n ( ξ ) ] = e i α [ g n ( ξ ) g n ( ξ ) ] = exp [ ( k θ y σ ) 2 ] .
I ( x , y ) ¯ = sinc 2 ( k θ x L 2 ) j a j 2 e ( j 2 π σ p ) 2 sinc 2 [ π p λ ( N + 1 ) ( θ y j λ p ) ] .

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